Arafat reportedly accepted the offer on Saturday although the resignation formalities have not yet been completed, Palestinian officials say.
But on hearing the news of the resignation offer, the Israeli government insisted on Saturday it would not deal with any Palestinian leadership shaped by Arafat.
"Israel will not accept a situation where control of the Palestinian Authority is returned to Arafat or anyone who does his bidding," said a statement from Israeli Prime Minister Ariel Sharon's office.
Abbas was reportedly dissatisfied that after 120 days as premier he has failed to achieve progress with the US-backed peace plan known as the road map.
The offer to resign comes after increasing violence in the Israeli-occupied West Bank and Gaza in the past fortnight.
The violence has left more than a dozen dead, scores injured, and the US-backed plan for Israeli-Palestinian peace efforts in tatters.
The Palestinians' chief negotiator, Saeb Erekat, told Aljazeera that Abbas blamed the United States and Israel for his decision, accusing the two of making statements supporting him without backing those words with practical measures.
Instead, Abbas believes the two had merely pursued Israeli interests, Erekat said.
Abbas blames Israel's continuing campaign of assassinations and arrests of Palestinian resistance leaders, despite a 29 June cease-fire they had issued, for undermining his attempts to achieve progress with the road map.
And the US administration had, in Abbas' view, done little to make Israel comply with the plan.
Differences with Arafat
Abbas also cited differences with Arafat, from whom the prime minister has unsuccessfully been trying to win more power. The prime minister had called on the council on Thursday to back him in his attempt to get more authority or sack him.
The premier had sought more power to carry out further reforms and rein in the resistance - measures required by the US-backed plan.
Palestinians protested forcefully
against the premier on Thursday
In addition, Erekat said Abbas had been dismayed by a Palestinian protest that targeted his supporters in the legislative council on Thursday. Abbas accused some protesters of trying to insult and attack him.
Aljazeera's correspondent said the dispute between the two leaders and its role in Abbas' resignation offer was a serious blow to the efforts of mediators, including recent attempts by Egypt, to bridge the gap between the president and prime minister.
The premier's departure could destroy the US-backed plan for peace with Israel, already reeling from the collapse of the 29 June cease-fire by resistance groups at the end of August and a Israeli military campaign to assassinate their leaders.
The main Palestinian resistance group Hamas, which opposed the appointment of Abbas, reacted unexpectedly to the news of his apparent resignation.
"I believe that this step is against the interests of Palestinians, as the establishment of this kind of government was against the interests of the Palestinians," said spokesman Abd al-Aziz al-Rantissi.
"The establishment (of the government) was the result of Israeli and American pressure, and the resignation now is because of pressure. So this...gives (Israeli Prime Minister Ariel) Sharon a chance to practise terror against Palestinians.
"I believed from the beginning of the first steps taken by Abu Mazin (Mahmud Abbas) as prime minister that he would fail because the Israelis would give nothing to the Palestinians."